If a, b and c are all non-zero and 1+a1111+b1111+c=0, then prove that 1a+1b+1c+1=0

If a, b and c are all non-zero and $|\begin{matrix} 1 + a & 1 & 1 \\ 1 & 1 + b & 1 \\ 1 & 1 & 1 + c \end{matrix}| = 0$ , then prove that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + 1 = 0$

Match the items on the right to the items on the left.